PDEs- D'Alembert Solution of the Wave Equation

In summary, the conversation is about a question from a linear partial differential equations assignment that involves finding the boundaries of an integral. The student is having trouble understanding how to determine the boundaries and is asking for clarification on a specific example.
  • #1
Roni1985
201
0

Homework Statement


The question is here:
http://ocw.mit.edu/courses/mathemat...ions-fall-2006/assignments/probwave1solns.pdf

It's a long question and I figured attaching the link here would be better.

I need help with the question on page 4.
when

Ut(x,0)=g(x)
and U(x,0)=0

Homework Equations



The only relevant equation is the integral of Integral[ g(s)ds,s,x-t,x+t]

The Attempt at a Solution



I tried to look at the solutions but I don't understand how to find the boundaries of the integral from the regions.

for example,
to be in region 2, the solution says that x+t>1 and -1<x+t<1
why is that ?
how was this derived ?Thanks,
Roni.
 
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  • #2
still can't figure this out :\
 

1. What is the D'Alembert solution of the wave equation?

The D'Alembert solution is a method for solving the wave equation, which describes how waves propagate through a medium. It involves splitting the equation into two parts, one representing the forward-moving wave and the other representing the backward-moving wave.

2. How is the D'Alembert solution derived?

The D'Alembert solution is derived by applying the method of separation of variables to the wave equation. This involves assuming a solution of the form u(x,t) = X(x)T(t) and then solving for the individual functions X(x) and T(t).

3. What are the assumptions made in the D'Alembert solution?

The D'Alembert solution assumes that the wave equation is linear, homogeneous, and has constant coefficients. It also assumes that the medium through which the wave is propagating is homogeneous and isotropic.

4. What are the advantages of using the D'Alembert solution?

The D'Alembert solution is advantageous because it provides a general solution to the wave equation that can be applied to a wide range of problems. It also allows for the superposition of multiple waves, making it useful for studying complex wave phenomena.

5. Are there any limitations to the D'Alembert solution?

Yes, the D'Alembert solution is limited to linear, homogeneous, and isotropic wave equations. It also assumes that the initial conditions are specified for the entire domain, which may not always be the case in real-world scenarios.

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